Preheat the oven to 350 degrees F (175 degrees C). Grease the sides of three 9-inch round cake pans; line bottoms with parchment paper.

Beat butter with sugar in a large bowl using an electric mixer until fluffy. Add eggs, 1 at a time, beating well after each. Mix in vanilla extract. Combine flour, baking powder, and baking soda. Stir into butter mixture alternately with buttermilk, making 3 additions of flour mixture and 2 of buttermilk.

Divide batter among 3 bowls. Tint one bowl red, one yellow, and one blue. Scrape each into one of the prepared pans and smooth the tops.

Bake in the preheated oven until a toothpick inserted into the centers comes out clean, about 30 minutes. Cool cakes on a wire rack for 15 minutes. Run a table knife around the edges to loosen. Turn out onto a cooling rack. Peel off parchment paper and let cool for 30 minutes before frosting.

Beat unsalted butter and salt together in a large bowl until smooth. Beat in confectioners' sugar alternately with cream, making 3 additions of sugar and 2 of cream. Mix in vanilla extract. Divide butter icing among 3 bowls. Tint one orange, one green, and the other purple.

Place red cake layer on flat serving plate; spread top with orange icing. Top with yellow layer; spread green icing on top. Add the blue layer and cover it with purple icing. Refrigerate for 30 minutes or until icing is firm.

Place corn syrup in a saucepan and bring to a boil. Meanwhile, beat egg whites and cream of tartar together until frothy in a heat-proof bowl set over a pan of simmering water. Beat in the corn syrup in a thin, steady stream until stiff peaks form. Remove from heat and beat in vanilla. Top cake with fluffy icing.